# Introduction to Fluid Dynamics

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2 Aim To provide a survey of basic concepts in fluid dynamics as a preliminary to the study of dynamical meteorology. Based on a more extensive course of lectures prepared by Professor B. R. Morton of Monash University, Australia. A useful reference is: D. J. Acheson Description of fluid flow The description of a fluid flow requires a specification or determination of the velocity field, i.e. a specification of the fluid velocity at every point in the region. u = u( xt, ) In general, this will define a vector field of position and time. Steady flow occurs when u is independent of time - i.e. u / t 0 Otherwise the flow is unsteady.

3 Streamlines Streamlines - lines which at a given instant are everywhere in the direction of the velocity (analogous to electric or magnetic field lines). In steady flow the streamlines are independent of time, but the velocity can vary in magnitude along a streamline (as in flow through a constriction in a pipe). fast slow constriction slow

4 Streamlines Particle paths Particle paths are lines traced out by "marked" particles as time evolves. In steady flow particle paths are identical to streamlines. In unsteady flow they are different, and sometimes very different. Particle paths are visualized in the laboratory using small floating particles of the same density as the fluid. Sometimes they are referred to as trajectories.

5 Filament lines or streaklines Filament lines or streaklines are traced out over time by all particles passing through a given point. They may be visualized, for example, using a hypodermic needle and releasing a slow stream of dye. In steady flow these are streamlines. In unsteady flow they are neither streamlines nor particle paths. Streaklines

6 Equations for streamlines The streamline through the point P, say (x,y,z), has the direction of u = u(x,y,z). streamline P Q u = u(x,y,z) u = u(x + δx, y + δy, z + δz) δx = u δt, δy = vδt, δz = wδt dx u dy = = v dz w Parametric representation of the streamlines dx dy dz = dt, = dt, = dt u v w

7 Example: Find the streamlines for the velocity field u = ( Ωy, Ωx, 0), where Ω is a constant Solution dx dy dz = = Ωy Ωx 0 Ω (xdx + ydy) = 0 dz = x + y = Γ( z) Γ an arbitrary function z = constant Streamlines are circles x 2 + y 2 = c 2, on planes z = constant Fluids as continuous media Fluids are molecular in nature, but they can be treated as continuous media for most practical purposes. The exception is rarefied gases.

8 Compressibility Real fluids generally show some compressibility defined as 1dρ κ = = ρ dp changes in density per unit change in pressure density At normal atmospheric flow speeds, the compressibility of air is a relative small effect and for liquids it is generally negligible. The exception is rarefied gases. Note that sound waves owe their existence to compressibility effects as do "supersonic bangs", produced by aircraft flying faster than sound. Incompressible fluids For many purposes it is accurate to assume our fluids are incompressible, i.e. they suffer no change in density with pressure For the present we shall assume also that they are homogeneous density ρ = constant

9 Friction in solids When one solid body slides over another, frictional forces act between them to reduce the relative motion. Friction acts also when layers of fluid flow over one another. When two solid bodies are in contact (more precisely when there is a normal force acting between them) at rest, there is a threshold tangential force below which relative motion will not occur. It is called the limiting friction. Example: a solid body resting on a flat surface under the action of gravity. N normal force on body T tangential force F frictional force on body N F tangential force on surface normal force on surface As T is increased from zero, F = T until T = µn. The coefficient of limiting friction, µ depends on the degree of roughness between the surface. For T > µn, the body will overcome the frictional force and accelerate.

10 Fluids compared with solids A distinguishing characteristic of most fluids in their inability to support tangential stresses between layers without motion occurring; i.e. there is no analogue of limiting friction. Exception: certain types of so-called visco-elastic fluids e.g. paint. A visco-elastic fluid A mixture of water and corn starch, when placed on a flat surface, flows as a thick, viscous fluid. However, when the mixture is rapidly disturbed, it appears to fracture and behave more like a solid.

11 Friction in fluids Fluid friction is characterized by viscosity which is a measure of the magnitude of tangential frictional forces in flows with velocity gradients. Viscous forces are important in many flows, but least important in flow past "streamlined" bodies. We shall be concerned mainly with inviscid flows where friction is not important. It is essential to acquire some idea of the sort of flow in which friction may be neglected without completely misrepresenting the behaviour. Its neglect is risky! Viscosity Water Silicone oil 10,000 times more viscous

12 Incompressible flow Consider a stream tube - an element of fluid bounded by a "tube of streamlines. S 1 S 2 mass flux = ρv 1 S 1 ( = mass flow per unit time) vs = constant mass flux = ρv 2 S 2 In steady flow, no fluid can cross the walls of the tube (they are everywhere in the direction of flow). In the limit, for stream tubes of small cross-section, vs = constant along an elementary stream tube. Where streamlines contract the velocity increases, where they expand it decreases. The streamline pattern contains a great deal of information about the velocity distribution. All vector fields with the property that (vector magnitude) * (area of tube) remains constant along a tube are called solenoidal. The velocity field for an incompressible fluid is solenoidal.

13 Conservation of mass - the continuity equation Apply the divergence theorem n u udv = u n ds V S V S to an arbitrarily chosen volume V with closed surface S. Assume: incompressible fluid and no mass sources or sinks within S. There can be neither continuing accumulation of fluid within V nor continuing loss. The net flux of fluid across the surface S must be zero, i.e., ds 0 S u n = dv 0 V u = This holds for an arbitrary volume V u = 0 This is the continuity equation for a homogeneous, incompressible fluid.

14 Equation of motion for an inviscid homogeneous fluid The equation of motion is an expression of Newton s second law of motion: mass acceleration = force To apply this law we must focus our attention on a particular element of fluid. We consider a small rectangular element which at time t has vertex at P [ = (x, y, z)] and edges of length δx, δy, δz. z O y x The mass of this element is ρ δx δy δz. Assume ρ constant ρ is the fluid density = mass per unit volume

15 The velocity in the fluid, u = u(x, y, z, t) is a function both of position (x, y, z) and time t. From this we must derive a formula for the acceleration of the element of fluid which is changing its position with time. Example: Consider steady flow through a constriction in a pipe. Fluid elements must accelerate into the constriction as the streamlines close in and decelerate beyond as they open out again. Thus, in general, the acceleration of an element (i.e., the rate-of-change of u with time for that element) includes a rate-of-change at a fixed position u/ t plus a change associated with its change of position with time. I will derive an expression for the total rate-of-change shortly.

16 Forces acting on the fluid element The forces acting on the elements δx, δy, δz consist of: (i) body forces, which are forces per unit mass acting throughout the fluid because of external causes, such as the gravitational weight, and (ii) contact forces acting across the surface of the element from adjacent elements. These are discussed further below. Rate-of-change moving with the fluid Consider first the rate-of-change of a scalar property, for example the temperature of a fluid, T = T(x, y, z, t), following a fluid element. Suppose that an element of fluid moves from P [ = (x, y, z)] at time t to Q [ = (x + x, y + y, z + z)] at time t + t. Q T = T(x + x, y + y, z + z) P u = u(x,y,z) T = T(x,y,z)

17 If we stay at a particular point (x 0, y 0, z 0 ), then T is effectively a function of t only. If we move with the fluid, T(x,y,z,t) is a function both of position (x,y,z) and time t. The total change in T between P and Q in time t is T = T T = T( x + x, y + y, z+ z, t + t) T( x, y, z, t) Q P The total rate of change of T moving with the fluid is T Tx lim lim ( + xy, + y, z+ z, t+ t ) T ( x, y, zt =, ) t 0 t t 0 t For small increments x, y, z, t we may use a Taylor expansion T T(x + x,y + y,z ++ z,t + t) T(x,y,z,t) + t + t T T T x + y z higher order terms in x, y, z, t. x + + P y z P P P The rate-of-change moving with the fluid element T T T T = lim t x y z / t t t x y z T T T T = + u + v + w, since higher order t x y z terms 0. Note that u = dx/dt, v = dy/dt, w = dz/dt, where r = r(t) is the coordinate vector of the moving fluid element.

18 Total rate-of-change moving with the fluid To emphasize that we mean the total rate of change moving with the fluid we write DT Dt T u T = + + v T + w T t x y z the local rate-of-change with time at a fixed position u T x + v T w T T y + z = u is the advective rate-of-change associated with the movement of the fluid element Imagine that one is flying in an aeroplane that is moving with velocity c(t) = (dx/dt, dy/dt, dz/dt) and that one is measuring the air temperature with a thermometer mounted on the aeroplane. T T air (t) x x = x(t) = ct If the air temperature changes both with space and time, the rate-of-change of temperature that we would measure from the aeroplane would be dt T = + c T dt t

19 dt dt T = + c T t The rate at which the temperature is varying locally; i.e., at a fixed point in space. The rate-of-change that we observe on account of our motion through a spatiallyvarying temperature field. Suppose that we move through the air at a speed exactly equal to the local flow speed u, i.e., we move with an air parcel as in a balloon. x T u T balloon (t) x = x(t) = ut The rate-of-change of any quantity related to the air parcel, for example its temperature or its x-component of velocity is given by t + u operating on the quantity in question.

20 The total derivative We call t + u the total derivative and often use the notation D/Dt for it. Thus the x-component of acceleration of the fluid parcel is Du Dt u = + u u t while the rate at which its potential temperature changes is expressed by Dθ θ = + u θ Dt t Case of potential temperature In many situations, θ is conserved following a fluid parcel, i.e., D θ = 0 Dt θ t = u θ The rate-of-change of potential temperature at a point is due entirely to advection. The change occurs solely because fluid parcels arriving at the point come from a place where the potential temperature is different.

21 Example Suppose that there is a uniform potential temperature gradient of -1 C/100 km between Munich and Frankfurt, i.e. the air temperature in Frankfurt is cooler. cold U warm Frankfurt Munich If the wind blows directly from Frankfurt to Munich, the potential temperature in Munich will fall steadily at a rate proportional to the wind speed and to the temperature gradient. If the air temperature in Frankfurt is higher than in Munich, then the potential temperature in Munich will rise. cold air advection θ t θ = U <0 x cold Frankfurt U warm Munich x warm air advection θ θ = U >0 t x warm Frankfurt U cold Munich x

22 Example DF Dt F T Show that = + ( u ) represents the total rate-of-change of any vector field F moving with the fluid velocity (velocity field u), and in particular that the acceleration (or total change in u moving with the fluid) is F Solution Du Dt u = + ( u ) u t The previous result for the rate-of-change of a scalar field can be applied to each of the component of F, or to each of the velocity components (u,v,w) and these results follow at once. Example Show that Dr Dt = u Solution Dr Dt r = + ( u ) r t = 0+ u + v + w (x,y,z) x y z = (u,v,w) as x, y, z, t are independent variables.

23 Question Are the two x-components in rectangular Cartesian coordinates, ( ) x 1 2 u u and ( 2 u ) the same or different? x Note that 1 2 ( ) ( 2 ) u u = u u ω

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